Slow-Time Ambiguity Function Shaping With Spectral Coexistence for Cognitive Radar

In the spectrum-congested environment, weak target obscuration can be addressed by simultaneously shaping the ambiguity function and waveform spectrum based on cognitive radar a priori information. In this article, we investigate joint waveform design to suppress slow-time ambiguity function (STAF) disturbance power and overlapping frequency band energy. Thus, a novel design criterion involving minimizing the interested range-Doppler blocks of STAF and energy spectrum density stopband is constructed, which is subject to the waveform energy and peak-to-average ratio constraints. To cope with the resulting complex quartic optimization problem, a waveform design approach is proposed which utilizes the iterative sequential quartic optimization algorithm framework to obtain a closed-form solution at each iteration. Finally, the designed waveforms can suppress the interested range-Doppler block level while satisfying the spectral coexistence requirements. Numerical simulation results verify that the proposed method has higher STAF disturbance suppression performance than state-of-the-art methods. Meanwhile, this method also possesses the ability to reject narrow-band spectrum interference.

Abstract-In the spectrum-congested environment, weak target obscuration can be addressed by simultaneously shaping the ambiguity function and waveform spectrum based on cognitive radar a priori information.In this article, we investigate joint waveform design to suppress slow-time ambiguity function (STAF) disturbance power and overlapping frequency band energy.Thus, a novel design criterion involving minimizing the interested range-Doppler blocks of STAF and energy spectrum density stopband is constructed, which is subject to the waveform energy and peakto-average ratio constraints.To cope with the resulting complex quartic optimization problem, a waveform design approach is proposed which utilizes the iterative sequential quartic optimization algorithm framework to obtain a closed-form solution at each iteration.Finally, the designed waveforms can suppress the interested range-Doppler block level while satisfying the spectral coexistence requirements.Numerical simulation results verify that the proposed method has higher STAF disturbance suppression performance than state-of-the-art methods.Meanwhile, this method also possesses the ability to reject narrow-band spectrum interference.

I. INTRODUCTION
A MBIGUITY function is an important evaluation metric in radar systems, which reflects the range and Doppler resolution, as well as its own sidelobe disturbance power.It represents the range-Doppler response of the radar transmit waveform with time delay and Doppler shift to the matched filter [1].Essentially, the shape of ambiguity function is determined by the transmit waveform.Therefore, it is possible to implement ambiguity function shaping by waveform design for different environments and requirements to enhance the detection performance of interested targets [2], [3], [4], [5], [6], [7], [8], [9], [10].
It is well known that the integrated sidelobe level (ISL) of the ambiguity function is fixed.Therefore, an attempt to suppress all sidelobe level is unrealistic.Fortunately, it is possible to utilize the prior information provided by cognitive radar, e.g., target region, Doppler frequency range, to suppress the sidelobe levels of interest in the ambiguity function.Such an approach breaks through the limitation of energy fixation of the ambiguity function and helps to improve the target detection.For this reason, local ambiguity function (LAF) assignment has aroused extensive attention [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].
Depending on how the waveform is encoded, ambiguity function can be divided into fast-time ambiguity function and slow-time ambiguity function (STAF).Fast time is defined as the time scale corresponding to a single pulse.Slow time, on the other hand, is defined as the time scale corresponding to multiple pulses.Compared to fast time, slow time has a certain pulse repetition interval and the sampling time is slower than fast time.Therefore, the ambiguity function can be shaped by controlling the weighted integrated sidelobe level (WISL) of a single pulse in the fast-time dimension or by controlling the disturbance power of multiple pulses in the slow-time dimension, respectively.
Fast-time ambiguity function shaping refers to the direct optimization of waveform classic ambiguity function, especially its zero Doppler profile, which has been widely studied by many scholars [4], [12], [13], [14], [15].There are two specific approaches, one is to directly minimize the WISL, and the other is to minimize the mean square error of the ambiguity function with respect to the desired template, i.e., to let the ambiguity function approximate the given template as closely as possible [9], [16], [17], [19], [20].
STAF shaping can be regarded as designing slices of fast-time ambiguity functions at zero time delay and pulse repetition interval multiples.Such problems have attracted the attention of many scholars.Maximum block improvement (MBI) method is widely applicable to address polynomial optimization [19], involving the solution of linear and quadratic subroutines.Leveraging the MBI method, Aubry [9] solved the quartic optimization problem of STAF shaping under constant modulus constraint.In [16], Aldayel et al. proposed an efficient adaptive sequential refinement method to tackle nonconvex fourth-order problem exploiting convex quadratic programming (QP) sequences while updating the cost function and constant modulus constraint in each iteration.Compared to the MBI algorithm, this method shows a significant improvement in running time.Based on the coordinate descent methods, the authors design an optimization method to solve the quartic problem [17].This method optimizes the objective function in the form of a quadratic function and solves the resulting problem element by element using the coordinate descent method.Unlike [9], [16], [19], the constant modulus constraint is further extended to an energy constraint and a peak-to-average ratio (PAR) constraint [17], which contributes to the optimization performance.However, these studies only consider ambiguity function shaping and ignored the problem of scarce spectrum resources nowadays.
Therefore, spectral coexistence has become particularly important to avoid the interference caused by spectral overlap and has been studied by many researchers [21], [22], [23], [24], [25], [26].A shape method (SHAPE) is proposed to design waveforms with arbitrary spectrum [27], and so is [21], all of which ignore the ambiguity function properties of the waveform.In [28], [29], and [30], the combined optimization of autocorrelation and waveform spectrum shaping is investigated considering the spectral coexistence in complicated electromagnetic environments.Nevertheless, these works neglect the Doppler effect and only suppressed the range sidelobe level on zero Doppler.To this end, an iterative optimization method is developed to achieve radar ambiguity function shaping and spectrum sharing by suppressing the fast-time ambiguity function WISL while also minimizing the spectral stopband energy [13].However, there is no related work on STAF shaping with spectral coexistence.Moreover, most of the works consider the constant modulus constraint, which will limit the ambiguity function performance, although it can guarantee the operation of nonlinear amplifiers in saturation.The energy and PAR constraints are an extension of the constant modulus constraint.In other words, the constant modulus constraint is a special case of the energy and PAR constraints.Therefore, the waveform design problem considering the energy constraint and PAR constraint is worthy to investigate.
In this article, we address the joint shaping of the STAF and energy spectrum density (ESD).A novel design criterion involving minimizing the integrated range-Doppler bins power and the stopband energy of waveform spectrum is established.In addition, the energy constraint and PAR constraint are bound to the transmit waveform.To tackle the obtained nonconvex quartic problem, a waveform design method is developed based on the iterative sequential quartic optimization (ISQO) framework.The objective function is first converted into a quadratic convex optimization problem by solving for local eigenvalues in each iteration, and further transformed into a linear optimization problem with closed-form solutions.Furthermore, the SQUAREM acceleration algorithm is embedded to speed up the convergence of proposed method.The designed waveform can implement simultaneous shaping of STAF and ESD.Numerical simulations verify that the proposed method outperforms state-of-the-art approaches in terms of STAF interference power suppression.In addition, we also give the selection of weighting factors that contribute to the optimized performance.
The rest of this article is organized as follows.In Section II, a joint design criterion is constructed and an optimization problem is obtained by imposing constraints on it.A waveform design method based on the ISQO framework is developed to solve the above optimization problem in Section III.Then, we verify the performance of our method by numerical simulation results in Section IV.Finally, Section V concludes the article.
Notation: Italicized lowercase letters usually denote scalars.Bold lowercase refers to column vectors and bold uppercase letters means column matrices.The letter j represents the imaginary unit.{•} T and {•} H denote the transpose operation and conjugate transpose operation, respectively.| • | is the operation of taking absolute value, and • denotes Euclidean norm.
{•} represents the real part.diag(•) means the operation of forming a diagonal matrix from vector elements as diagonal elements.eig(•) indicates the eigenvalue of matrix.represents the Hadamard product.I N is the unit matrix of dimension N × N .A 0 represents that matrix A is positive semidefinite, where A ∈ H N .E{•} expresses the operation that takes the expectation.C N stands for a complex vector space of size N and H N is N × N Hermitian matrices.

II. PROBLEM FORMULATION
STAF shaping is aimed at suppressing some range-Doppler response of strong echoes, which improves the detection performance of weak targets.As for waveform, spectral shaping is to reduce the interference from other signals in the overlapping frequency bands.To optimize the detection performance of weak targets under spectral coexistence, a design criterion for simultaneous optimization of the STAF and the waveform ESD under energy and PAR constraints is constructed by weighting factors in this section.

A. STAF Shaping
Consider a monostatic radar system transmit a slow-time coded pulse with N -coherent burst.The radar transmitting pulse is expressed as s = [s(0), s(1), . . ., s(N − 1)] T ∈ C N , and the radar receiving pulse is denoted as v = [v(0), v(1), . . ., v(N − 1)] T ∈ C N .By resorting to rangeazimuth cell, the observed pulses v can be represented as where α T denotes the channel propagation and scattering effects of the target in the range-azimuth bin.p( where ν d T is one Doppler block after Doppler normalization.d is the vector of interfering echo samples, and n is the filtered noise vector with E[n] = 0 and E[nn H ] = σ 2 n I N .According to [16], the vector d expresses the return values from different N t interference scatterers located at different range-azimuth bins, which can be denoted as where ρ i is the echo complex amplitude, r i denotes the range direction, N t is the total number of interfering scatters, and v d i is the normalized Doppler frequency.J r i is a shift matrix with N × N dimension, which is defined as (3) The matched filter is s p(ν d T ).By combining (1) and ( 2), the output of the matched filter to the echo is given by where the last two terms present the disturbance to the target detection.Let E[|ρ i | 2 ] = δ 2 i denote the average power of the ith interference scattering.Then, the disturbance power is expressed as where g s (r, v h ) is the ambiguity function of transmitting signal s, denoted as where Generally, the constant modulus constraint or energy invariance constraint is imposed on the transmitting waveform.In this case, s 2 = N , which means that the noise term σ 2 n s 2 is a fixed value after matched filtering.Since the disturbance caused by noise cannot be changed by waveform optimization, the noise term can be neglected when designing the waveform.According to [16], the disturbance power can be cast as where p(r, h) represents the disturbance power at the scattering point, which lies at the range-Doppler bin (r, h).The normalized Doppler range is 6) with ( 7), we can obtain where (r, h) denotes a range-Doppler bin and each i corresponds to a (r, h) pair.Hence, C i is denoted as then Υ(s) is expressed as According to the expression of Υ(s), it is evident that Υ(s) is a N × N Hermitian matrix.Hence, it satisfies the following equation: From the definition of Φ 1 (s), it follows that the fact Φ 1 (s) ≥ 0. Therefore, we can determine that Υ(s) is a semipositive definite matrix and its eigenvalue is greater than or equal to 0.

B. ESD Shaping
The frequency stopband refers to the frequency band where the radar signal overlaps with other signals, i.e., the frequency where we expect to suppress energy.If there are N k stopbands, it is denoted as where f h1 and f h2 represent the lower and upper bounds of the hth stopband, respectively.According to [23], the energy of all frequency stopbands can be expressed as where R = N k h=1 c h R h c .c h denotes the weight of each stopband, and R h c can be obtained by

C. Design Criteria
To improve the target detection performance [i.e., matching the signal-to-interference ratio (SIR) of the filtered output] in a spectrum-congested environment, we construct a new objective function while considering the energy and the PAR constraints.The final problem is expressed as where λ ∈ [0, 1] is a weighting factor.The larger λ means the greater the importance of STAF shaping and the weaker the suppression of spectral stopband, and vice versa.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
permissible modulus of the PAR constraint which is defined as Specifically, when γ = 1, the constraint |s n | ≤ γ degrades to constant modulus constraint.To facilitate problem solving, minimization can be equivalently converted to maximization, and then the problem (15) can be rewritten as It can be found that the objective function is a quartic optimization problem.At the same time, the energy and PAR constraint makes the optimization problem difficult to handle.Next, we devise a waveform design method based on ISQO framework that can give a closed-form solution by solving a linear problem at each iteration.

III. SOLVING ALGORITHM
In this section, we develop a waveform design method leveraging the ISQO paradigm, which can obtain transmitting waveforms that satisfy the PAR and energy constraints.Besides, a closed-form solution of the convex optimization subproblem can be given at each iteration, guaranteeing that the objective function rises monotonically.
The objective function of P 1 is a combination of a quartic function and a quadratic function, and the PAR constraint causes P 1 a complex nonconvex optimization problem.To deal with this problem, a solution is derived by approximating the initial quartic function with a quadratic optimization.Proposition 3.1: P 1 can be solved by approximating it to the following quadratic optimization problem: where To ensure (λ q I N − λΥ(s (t) )) 0, λ q is required to satisfy Combined with the knowledge of matrix theory, λ u that meet the requirement can be set as Obviously, C i ss H C H i 0. We can also derive its rank is 1 [13].Due to the fact that Υ(s (t) ) is composed of multiple Hermitian matrices superimposed, there is the following inequality Algorithm 1: ISQO algorithm.
Proof 1: In order to be able to iterate, we first need to initialize transmitting waveform s (1) .Assuming that the transmitting waveform at the tth iteration is s (t) , we can optimize (18) by fixing Υ(s (t) ).As a result, we can obtain the optimal solution s (t+1) for one iteration.In this case, the following relationship holds: q s (t+1) ,s (t) ≥ q s (t) ,s (t)  (23) Taking into account the previous analysis, it can be deduced that the following inequality relation is valid: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The relation s (t) = s (t+1) = s * will be established when t trends to infinity.Therefore, the objective function value can be demonstrated to be monotonically increasing and has an upper limit as t approaches infinity.Furthermore, P 2 is a quadratic convex function, which can be converted into a linear function by a first-order Taylor expansion, i.e., where Hence, a closed-form solution can be obtained [31].Note that the constraints are not yet applied.Without loss of generality, we arrange v(s (t) ) in order from largest to smallest, i.e., where the number of nonzero elements is k.v n (s (t) ) is the nth element of v(s (t) ).Thus, the designed waveform is where arg(•) indicates the phase extraction operation.The amplitude |s n (t+1) | of transmitting waveform can be obtained as follows: where ρ is the scaling factor to adjust the waveform amplitude to meet the energy and PAR constraints, and it is to satisfy and Hence, the bisection method can be utilized to solve for ρ.The optimal solution of transmitting waveform can be subsequently obtained in one iteration.For convenience, s (t+1) is expressed as To improve the speed of the algorithm convergence, we inserted the SQUAREM accelerator [32].Algorithm 1 shows the overall framework of ISQO.In particular, the ability of the objective function value Γ(s) to decrease monotonically is achieved by step 8. Therefore, the objective function can be maximized by multiple iterations, i.e., the minimization of the STAF perturbation power and ESD stopband.

IV. NUMERICAL SIMULATIONS
This section is dedicated to evaluate the performance of the proposed method regarding the disturbance power of STAF and the frequency stopband energy.We compared the proposed method with the following three state-of-the-art approaches for STAF assignment: 1) the coordinate iteration for ambiguity function iterative shaping (CIAFIS) [17]; 2) the quartic gradient descent (QGD) [11]; 3) the majorized iteration for ambiguity function iterative shaping (MIAFIS) [17].
Consistent with previous work [11], [17], we adopt the same parameter settings: The Doppler frequency is divided into 50 blocks (N v = 50), which generates a discrete normalized Doppler frequency axis v h = − 1 2 + h N v , h = 0, . . ., 49.The SIR is employed to evaluate the numerical results of all algorithms and is defined as Following the previous work, two scenarios that have already been studied in [11] are examined.In addition, we explore the case of simultaneous STAF disturbance and narrow-band spectral interference.All scenarios assume that the disturbance power is equal between distance Doppler blocks.Initial waveforms are randomly generated.The simulation results in this article are obtained using MATLAB R2020b.Running device is a computer with Windows 10 operating system.Its CPU is Intel(R) Core(TM) i5-10400, and the size of its RAM is 8 GB.

A. Scenario 1
Fig. 1 shows the expected range-Doppler power distribution for STAF with the corresponding p(r, h) shown below Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.This scenario assumes that there is only the disturbance of range-Doppler blocks of STAF and no narrow-band spectral interference.Accordingly, the weighting factor λ is 1.To be fair, γ is set to 1 so that the PAR constraint becomes a unit modulus constraint.The length N of the transmitting waveform is 50.Fig. 2 shows the STAFs designed by the comparison methods and the proposed method.As the figure shows, all four methods appear notches in the red region shown in Fig. 1, which means that the interference power of the interested STAF range-Doppler block is suppressed.In particular, the proposed method has the lowest disturbance power, down to −250 dB.Compared to the MIAFIS method that also employs the MM framework, the proposed method reduces the number of surrogate function constructions and nonconvex eigenvalue solving.Therefore, the optimization performance of the proposed method is superior to that of the MIAFIS method.
To demonstrate the suppression effect visually, we depict the profiles of the ambiguity function cut along r = 1, 2, 3, 4, as shown in Fig. 3.As we expected, notches appear in the corresponding Doppler bins and the proposed method has the lowest sidelobe level, almost reaching −300 dB.There is no depression in the range-Doppler block without disturbance.

B. Scenario 2
In this scenario, the STAF interference distribution p(r, h) shown in Fig. 4 is expressed as Unlike Scenario 1, the interference region increases the first ten range cells at zero Doppler, which is commonly found in remote sensing applications, i.e., scattering points with different time delays share the same Doppler frequency.The weighting factor λ and the PAR constraint γ are both 1.The difference is that the waveform length N is 100.
In this case, the ambiguity functions of the waveforms designed by the CIAFIS, QGD, MIAFIS, and proposed methods are shown in Fig. 5.It is clear that, as well as notches in the red region, the ambiguity functions also have notches in the first ten range cells at the zero Doppler position.Moreover, the proposed method has the deepest notches compared to the comparison methods.In addition, we calculated the SIR of the CIAFIS, QGD, MI-AFIS, and proposed methods at different waveform lengths according to (35) as shown in Fig. 7. Obviously, the SIR improves as the waveform length increases.Furthermore, compared to the comparison methods, the proposed method has the highest SIR, attaining 286.49dB when N = 100.
Therefore, based on the results of Scenario 1 and Scenario 2, it can be concluded that the proposed method performs better than the three comparison methods in suppressing disturbance power of the STAF.

C. Scenario 3
The joint objective function is constructed in this article to achieve both ambiguity function assignment and frequency spectrum sharing.
For this scenario, we set the interference map p(r, h) to be consistent with Scenario 1, as well as the presence of a narrow-band spectrum interference at the location [0.3, 0.4] of the normalized frequency spectrum.Apparently, both λ and γ affect the optimization performance of ambiguity function and ESD.λ indicates the importance of the proposed method to suppress the interference power of ambiguity function, which directly impacts the final optimization result.The PAR constraint is in a sense an extension of the strict constant modulus constraint, which increases the freedom of the waveform design.Accordingly, the optimization properties can also be improved relative to the constant modulus case.
First, we discuss the properties of the proposed method in terms of AF interference power suppression and waveform frequency spectrum shaping when λ is 0, 0.05, 0.5, 0.95 and 1. γ is 1 and waveform length N is 100.Fig. 8 shows the ESDs of design waveform and Fig. 9 depicts four profiles of STAF, which is cut along r = 1, 2, 3, and 4, respectively.
Obviously, the stopband energy of ESD decreases as the weight factor λ decreases, and reaches a minimum when λ is 0. However, we can find that the range-Doppler block level of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.interest is only as low as −300 dB at λ = 1, and the energy of interested range-Doppler blocks is maintained at −100 dB for λ = 0.05, 0.5 and 0.95 from Fig. 9.
To further explore the effect of weighting factor λ on ESD and STAF, we depict the average values of ESD stopbands and local STAF (LSTAF) at different λ (each point is the average of 50 random experiments), as shown in the solid line in Fig. 10.In Fig. 10, the left axis represents the average of ESD stopband and the right axis denotes the average of LSTAF.Obviously, except for endpoints 0 and 1, the weight factor λ has negligible effect on the LSTAF at 0.05-0.95,and the stopband of ESD will be smaller as the weight factor decreases.Therefore, λ can be set to a smaller value such as 0.05 when both LSTAF and frequency spectrum need to be optimized.In addition, we also investigate the properties of our method at different weight factors when PAR is 4 (γ is 2), as shown in the dashed line in Fig. 10.It is clear that the performance of both LSTAF and ESD is better when PAR is 4 than when PAR is 1.Therefore, when the waveform amplitude is not strictly required, the waveform performance can also be improved by appropriately increasing the PAR.
In summary, the proposed method is superior to the CIAFIS, QGD, and MIAFIS methods in terms of STAF interference suppression.Moreover, the proposed method can achieve the suppression of both STAF interference power and ESD stopband energy.In particular, we discuss the effects of weighting factor and PAR on the optimization performance and give the parameter selection when the range-Doppler block power disturbance and narrow-band spectral interference are present simultaneously.

D. Application of Designed Waveform
In this section, we provide the experiments of target detection to illustrate the application scenarios of the proposed method.In addition, the influence of interference power on weak target detection is investigated.The unoptimized and designed waveforms are applied to detect two high-speed targets in the presence or absence of a narrow-band interference emitted by a nearby electronic device, respectively.
Assume a cognitive radar with the working wavelength λ = 0.03 m, the sampling interval T s = 1 × 10 −6 s and the transmit signal bandwidth B = 1 MHz.Suppose a strong target and a weak target coexist in the detection scenario.The strong target with velocity v 1 = 600 m/s (normalized frequency , where c is the light speed and • denotes the rounding down).The weak target with velocity v 2 = 1425 m/s (normalized frequency f 2 = 0.095) is located at R 2 = 100.75km (range cell l 2 = 671).

1) No Narrow-Band Interference:
The nth element of the received echo y can be expressed as (38) where v(n) is a zero-mean circularly symmetric complex Gaussian random variable with variance σ 2 v ; a 1 and a 2 are the amplitudes for strong and weak target, respectively.
The range-velocity plane for the cross ambiguity function (CAF) can be obtained by the following equation [9]:  proposed method is obtained at λ = 1.Fig. 11 depicts the range-velocity planes of CAF for the unoptimized waveform and the designed waveforms obtained by MIAFIS and the proposed methods.Obviously, the unoptimized waveform can only detect the strong target and cannot detect the weak one.In contrast, MIAFIS and the proposed methods can detect the weak target well, which is attributed to the excellent ambiguity function shaping.
Fig. 12 shows profiles of the range-velocity plane, corresponding to the black dashed line in Fig. 11.The weak target levels are almost the same for both methods, but MIAFIS has one sidelobe in the weak target region, which may lead to inaccurate detection.Therefore, the proposed method performs better than the MIAFIS method in detecting weak targets.
2) Narrow-Band Interference: In this scenario, the nth element of the received echo y can be expressed as where p(n) = a 3 e j2πf 3 n is a narrow-band interference signal.We set f 3 = 0.12 and |a 3 | 2 = −10 dB.The designed waveform of the proposed method is obtained by setting λ = 0.05.Other parameters are consistent with the scenario "No narrow-band interference."When a strong narrow-band interference exists in the detection scene, the range-velocity planes of the unoptimized waveform, MIAFIS method and the proposed method are shown in Fig. 13.It can be seen that the unoptimized waveform and MIAFIS method can only detect the strong target, while the weak target will be missed due to interference.However, the proposed method can still detect the weak target, thanks to its ability to suppress narrow-band interference.
Fig. 14 presents the range-velocity profiles at the black dashed line of Fig. 13.In the region where the weak target may exist, MIAFIS cannot effectively suppress the narrow-band interference, thereby failing to detect the weak target.In contrast, the proposed method can simultaneously suppress the strong target's sidelobe and narrow-band interference.
3) Effect of Interference Power: To verify the ability of the designed waveforms against narrow-band interference, we carried out experiments under different interference power, i.e., |a 3 | 2 is set from −60 dB to 0 dB with the increment of 5 dB.The power of the strong target is −20 dB and that of the weak one is −50 dB.
To evaluate the effect of interference power on weak target detection, the signal to interference plus noise ratio (SINR) of the weak target region is defined as where Γ is the weak target region, (r w , f w ) denotes the location of the target, and K is the number of sidelobes in the weak target region.Fig. 15 depicts the variation of SINR Γ with interference power.result is average of 100 run results.It be seen that the SINR Γ remains basically unchanged when the interference power is lower than −20 dB.When the interference power is higher than −20 dB, the SINR Γ will gradually decrease with the increase of interference power.When the interference power is −10 dB (the target power is −50 dB), the SINR Γ Fig. 14.Range-velocity plane profiles of MIAFIS and proposed method under narrow-band interference.Fig. 15.SINR Γ in the weak target region varies with interference power.can maintain 4.32 dB, which indicates that the proposed method can suppress interference well and enhance weak target detection.
In conclusion, the above experimental results illustrate the superior performance of the proposed method in terms of antiinterference and weak target detection.

V. DISCUSSION
In this article, a waveform design method with joint optimization of waveform spectrum and STAF is proposed for cognitive radar.The experimental results verify the effectiveness of the proposed method, and the designed waveform can suppress the narrow-band interference and improve the detection performance of the weak target.
From the optimization problem model, it can be seen that there are three parameters that affect the optimization performance: weight factor λ, PAR, and waveform length N .Currently, there are no standard guidelines for the selection of the weight factor and PAR.However, we can select them according to the specific scenario by combining their effects on ESD and SIR.The details about the weight factor λ selection are as follows: 1) If there are only strong and weak targets in the detection scenario without narrow-band interference, the weight factor λ is selected as 1 to ensure the best target detection performance.2) If there are no weak targets in the detection scenario but there is a narrow-band interference signal, λ is chosen as 0 to guarantee the suppression of the narrow-band interference.3) If there are both strong and weak targets present in a detection scene with narrow-band interference, the selection of λ is complex and needs to be determined by the specific scenario.For example, in terms of Fig. 10, λ can be set as about 0.05 to tradeoff the detection of weak targets and the suppression of narrow-band interference well.For PAR, it can be seen from Fig. 10 that the larger the PAR, the better the optimization performance is.Generally, the PAR with a value of 1 can ensure that the nonlinear power amplifier works in a saturation state where the amplitude of the radar waveform is constant.Therefore, PAR can be increased appropriately as long as the power requirements are satisfied.
In addition, the waveform length N is the main determinant of the degrees of freedom in waveform design.Fig. 16 describes the effect of the waveform length on SIR.As depicted in this figure, the curve exhibits three distinct stages: 1) a steep increase, 2) a gradual increase, and 3) a plateau.The reasons for this trend are analyzed as follows.
1) In cases where the waveform length is short, typically below 60, the dominant factor affecting optimization performance is the degree of freedom in waveform design.
Increasing the waveform length allows for more design possibilities, resulting in a steep increase in the SIR.2) When the waveform length is relatively long, such as in the range of 60 to 100, the rate of improvement in the SIR with increasing waveform length slows down.This could be attributed to additional factors that limit optimization performance, including the approximate optimization function and waveform constraints.3) When the waveform length exceeds a certain value, typically greater than 100, the SIR tends to reach a plateau and remains relatively unchanged despite further increases in waveform length.In this situation, the dominant factors influencing waveform optimization are the approximate optimization problem and waveform constraints.Although the proposed method has superior capabilities in terms of antijamming and weak target detection, there still exists improvement space for target detection, such as the suppression of the peak sidelobe level of ambiguity function.In the future, we will introduce the peak sidelobe level constraints while optimizing the transmitted waveform to improve the detection performance of targets.

VI. CONCLUSION
In this article, the problem of the transmitting waveform STAF shaping for cognitive radar in a spectrally congested environment is addressed.A novel joint design criterion involving minimizing the STAF range-Doppler block power and ESD stopband energy is first established.In addition, the energy and PAR constraints are simultaneously imposed on the waveform.Based on the ISQO algorithm framework, a waveform design method for solving the resulting quartic problem is devised.During each iteration, the initial cost function is approximated as a quadratic optimization problem, which can be further converted into a solvable problem by the first-order Taylor expansion.In particular, by analyzing the effect of different weighting factors on the optimization results, we give parameter choices for the simultaneous shaping of the local STAF and ESD.Finally, the numerical simulation results indicate that the proposed method has superiority over the CIAFIS, QGD, and MIAFIS methods in STAF shaping.Additionally, the proposed method can suppress both the range-Doppler disturbance power and the spectral stopband energy.

Fig.
Fig. Desired power distribution of STAF for scenario 2 (only 25 range bins are shown).

Fig. 6 (
a) and (b) are the partial profiles of Fig. 5. Fig. 6(a) and (b) represents the profiles of ambiguity functions along h = 25, r = 1, r = 2, r = 3 cuts, respectively.Consistent with the results presented in Fig. 5, Fig. 6(a) (autocorrelation function) has suppressed sidelobe levels within ten range cells of the mainlobe; Fig. 6(b)-(d) all present a depression at the zero Doppler frequency location.

Fig. 10 .
Fig. 10.ESD and average of LSTAF versus λ or PAR.(Each value is the average of 50 random trials.).

Fig. 12 .
Fig. 12. Range-velocity plane profiles of MIAFIS and proposed method without narrow-band interference.